An estimate for the volume entropy of nonpositively curved graph manifolds

S. Buyalo ^{∗}

Abstract

Let M be a closed 3-dimensional graph manifold. We prove that h(g) > 1 for each geometrization g of M, where h(g) is the topological entropy of geodesic flow of g.

1 Introduction

Asymptotic geometry of a nonpositively curved (NPC for brevity) graph manifold is a complicated mixture of flat and hyperbolic parts which both nontrivially contribute in general picture. Recall that a NPC metric on a closed 3-dimensional graph manifold M recovers the JSJ-decomposition of M in the following sense. There is the unique (up to isotopy) minimal finite collection E of flat geodesically embedded tori and Klein bottles which are pairwise disjoint, and the metric completion of each connected component of the complement to E is a Seifert space called a block of M. Each block M_{v} is fibered over a 2-orbifold S_{v} with negative Euler characteristic, χ(S_{v}) < 0. Furthermore, the metric locally splitts as U × (−ε, ε) along the interior of each block, where U is a NPC surface, the splitting is compatible with the fibration, the fibers are closed geodesics, and the regular fibers have one and the same length l_{v} > 0 depending only on the block.

Since we are interested in asymptotic properties, which are certainly the same for any finite covering of M, we may assume for simplicity that M is orientable, the collection E consists of tori, and each block is a trivial S^{1}-bundle over a compact surface S_{v} with boundary, M_{v} = S_{v} × S^{1}. We also assume that the graph manifold structure of M is nontrivial, i.e. M itself is not a Seifert fibered space (though it may consist of one block).

The flat part of the asymptotic geometry of M studied in [BS], [CK], see also [HS]. Roughly speaking, it can be described by very special geodesic rays [0, ∞) → M, which terminate in no block, skip through separating tori e ∈ E almost tangently spending most of the time near tori and moreover this time rapidly increases with each step. Though the set of such rays is a negligible part all of the rays, it contains an important information about geometry of M: it was shown in [CK] how this information allows to recover (up to scaling) the marked length spectrum of closed geodesics on S_{v} and the fiber length for each block M_{v}.

∗Partially supported by RFFI Grants 99-01-00104 and 00-15-96024.

Here we study the hyperbolic part of the asymptotic geometry of M, assuming that the surface U from the local splitting U × (−ε, ε) above has the constant curvature K = −1. In other words, each block fibers over a hyperbolic orbifold (surface) S_{v}. A NPC metric on M satisfying this condition is called a geometrization of M (because the metric of each block is modelled on H^{2} ×ℝ). Note that any geometrization of M is only C^{1,1}-smooth being analytic along the interior of each block. It is known [L] that M admits a NPC metric if and only if it admits a geometrization. Necessary and sufficient topological conditions for M to carry a NPC metric were found in [BK].

The relevant metric invariant which measures hyperbolicity of a space is the volume entropy h. Let π : X → M be the universal covering, x_{0} ∈ X. Recall that h = h(X) is defined by

h = lim_{R→∞} \frac{1}{R} ln vol B_{R}(x_{0}),

where B_{R}(x) is the ball in X of radius R centered at x. It is well known [M] that the limit exists, h is independent of the choice of x_{0} and if M is NPC then h coincides with the topological entropy of geodesic flow of M. The volume entropy scales as l^{−1}, l is the length. Thus the choice of a geometrization g of M also serves as a normalization. The sectional curvatures of g satisfy −1 ≤ K ≤ 0. Hence h(g) ≤ 2 by comparison with H^{3}. Our main result is this.

Theorem 1.1. For any geometrization g of a graph manifold M we have h(g) > 1.

Remark 1.2. Though the universal covering X of M looks much more complicated than the model space H^{2} ×ℝ, even the estimate h(g) ≥ 1 = h(H^{2} ×ℝ) is not obvious and nontrivial: X contains no isometrically and geodesically embedded H^{2}, which would lead to h(g) ≥ 1; on the other hand, the attempt to compare X → H^{2} ×ℝ via exponential maps identifying some tangent spaces fails because the Jacobian of this map is > 1 at some points. Finally, the estimate from [BW] for the measure theoretic entropy of geodesic flow, which is always ≤ h(g), gives only π/4 < 1 as a lower bound for any geometrization g of M (if one ignores the fact that C^{1,1}-smoothness of g is not sufficient to apply this estimate).

To prove Theorem 1.1 we use the well known fact that h(g) coincides with the critical exponent of a Poincaré series

P(t) = ∑_{γ∈Γ} e^{−t|x_{0}−γx_{0}|},

where the fundamental group Γ = π_{1}(M) isometrically acts on X as the deck transformation group. Actually, instead of P we use a modified Poincaré series P_{W}, where summation is taken over a set W of walls in X. Our proof involves three ingredients: (i) a local estimate, which is technical and used in (ii); this estimate is obtained in section 2. (ii) An accumulating procedure, which is an inductive construction of appropriately choosen broken geodesics in X between the base point x_{0} and the walls from W; the choice of these paths is the key point of the proof. The outcome of the accumulating procedure is a generating set for P_{W} used in (iii); the procedure is described in sect. 3. (iii) A self-similarity type argument. This part of the proof uses a standard idea from self-similarity theory to show that P_{W} (h) diverges for some h > 1 using the generating set obtained in (ii). This is done in sect. 4.

Acknowledgment. The author is grateful to W. Ballmann for useful discussions of the topic of this note.

2 Local estimate

Let F be the universal covering of a compact hyperbolic surface S with geodesic boundary. We identify F with a convex subset F ⊂ H^{2} bounded by countably many disjoint geodesic line and fix o ∈ H^{2} \F. Let w_{0} be the boundary line of F closest to o, o_{0} ∈ w_{0} be the point on w_{0} closest to o, so that |o − o_{0}| = dist(o, F) =: l > 0.

Let A be the set of the boundary lines of F different from w_{0}. For each w ∈ A we denote by o_{w} ∈ w the point closest to o. Then the geodesic segment oo_{w} intersects w_{0} at some point t_{w}, and for l̃_{w} = |o − o_{w}| we have

l̃_{w} = l̃′_{w} + l′′_{w},

where l̃′_{w} = |o − t_{w}|, l′′_{w} = |t_{w} − o_{w}| (all distances are taken in H^{2}).

Next, we identify H^{2} with H^{2} ×0 ⊂ H^{2} ×ℝ, so that F becomes a subset of H^{2} ×ℝ, and we use for it the notations introduced above. Note that the point o_{0} is closest to o from the wall w_{0} × ℝ. We take a geodesic line σ ⊂ w_{0} × ℝ through o_{0} which is not horizontal, i.e., σ ≠ w_{0} × 0, and take s_{w} ∈ σ with |s_{w} − o_{0}| = |t_{w} − o_{0}|. Now, we put l′_{w} := |o − s_{w}| (the distance is taken in H^{2} ×ℝ), Δ_{w} := l̃′_{w} − l′_{w}.

In other words, we replace the distance l̃′_{w} between o and t_{w} is the hyperbolic plane H^{2} by the distance l′_{w} in H^{2} ×ℝ, which is shorter by comparison: the triangles oo_{0}t_{w} ⊂ H^{2} ×0, oo_{0}s_{w} ⊂ H^{2} ×ℝ both have the right angles at o_{0}, ∠(oo_{0}t_{w}) = \frac{π}{2} = ∠(oo_{0}s_{w}), the common side oo_{0} and equal sides |o_{0} − t_{w}| = |o_{0} − s_{w}|. Since oo_{0}t_{w} lies in the hyperbolic plane H^{2} ×0 but oo_{0}s_{w} not, we have Δ_{w} > 0 except the case t_{w} = o_{0} = s_{w}. Now, we want to estimate accumulation of the differences Δ_{w} from below. The precise statement is this.

Lemma 2.1. Given l_{0} > 0, α_{0} ∈ (0, π/2], there exists λ_{0} > 1, which depends only on l_{0}, α_{0} and the compact surface S, so that

λ(F, l, α) := e^{l} ∑_{w∈A} e^{Δ_{w}}e^{−l̃_{w}} ≥ λ_{0},

whenever l = dist(o, w_{0}) ≥ l_{0} and the angle α between the lines w_{0} × 0 and σ is at least α_{0}, α_{0} ≤ α ≤ π/2.

Proof. By well known formula of hyperbolic geometry we have e^{l} = (tan \frac{ψ}{4} )^{−1}, e^{−l̃_{w}} = tan \frac{ψ_{w}}{4} , where ψ, ψ_{w} are the angles under which w_{0} respectively w ∈ A are observed in H^{2} from o. The boundary at infinity ∂_{∞}F ⊂ ∂_{∞} H^{2} = S^{1} coincides with the limit set of π_{1}(S) represented in Iso(H^{2}) as a Fuchsian group of second kind. It is well known that the Hausdorff dimension of ∂_{∞}F (with respect to the angle metric) is < 1, in particular, the Lebesgue measure of ∂_{∞}F is zero. Thus ψ = ∑_{w∈A} ψ_{w}. Therefore, tan \frac{ψ}{4} ≤ ∑_{w∈A} tan \frac{ψ_{w}}{4} and

e^{l} ∑_{w∈A} e^{−l̃_{w}} = ∑_{w∈A} τ_{w} ≥ 1,

where τ_{w} = e^{l−l̃_{w}}. However, this sum, ∑_{w∈A} τ_{w}, can be close to 1 as much as we like (taking for instance l → ∞). As we noticed above, Δ_{w} > 0 unless s_{w} = t_{w}. So we always have λ(F, l, α) > 1. The point is that λ(F, l, α) is separated from 1 uniformly over all l ≥ l_{0}, α ≥ α_{0}.

Consider A_{0} ⊂ A consisting of all w ∈ A with |t_{w} − o_{0}| ≥ 1. Then Δ_{w} ≥ δ_{0} > 0 for all w ∈ A_{0}, where δ_{0} depends only on l_{0}, α_{0}. We claim that

∑_{w∈A_{0}} τ_{w} ≥ m_{0} > 0,  (1)

where m_{0} = m_{0}(F̃, l_{0}) is independent of l.

Assuming (1), we have

λ(F, l, α) = ∑_{w∈A} e^{Δ_{w}}τ_{w} ≥ ∑_{w∈A_{0}} e^{Δ_{w}}τ_{w} + ∑_{w∈A\A_{0}} τ_{w}
≥ e^{δ_{0}} ∑_{w∈A_{0}} τ_{w} + ∑_{w∈A\A_{0}} τ_{w}
= (e^{δ_{0}} − 1) ∑_{w∈A_{0}} τ_{w} + ∑_{w∈A} τ_{w} ≥ (e^{δ_{0}} − 1)m_{0} + 1 =: λ_{0} > 1.

It remains to prove (1). Let o′_{w} ∈ w be the point closest to o_{0} ∈ w_{0}. Then l̃_{w} ≤ |o − o′_{w}| ≤ l + |o_{0} − o′_{w}|, thus l̃_{w} − l ≤ dist(o_{0}, w). Hence τ_{w} ≥ e^{− dist(o_{0},w)} ≥ \frac{ψ̃_{w}}{4} , where w is observed from o_{0} under the angle ψ̃_{w}.

The Lebesgue measure class on ∂_{∞} H^{2} is independent of the choice of origin, thus ∑_{w∈A} ψ̃_{w} = π.

Since l ≥ l_{0}, for a sufficiently small m_{0} = m_{0}(S, l_{0}) > 0 the sectors S^{+}(m_{0}), S^{−}(m_{0}), defined below, intersect no w ∈ A \ A_{0}. Here is the definition of S^{±}(m_{0}). The common vertex o_{0} of S^{±}(m_{0}) divides the line w_{0} into two opposite rays w_{0}^{±}. The sectors S^{±}(m_{0}) ⊂ H^{2} are bounded by the rays w_{0}^{±}, s^{±}(m_{0}), where ∠_{o_{0}}(s^{±}(m_{0}), w_{0}^{±}) = 2m_{0}, and s^{±}(m_{0}) ∩ F ≠ ∅.

Therefore, it follows from ∑_{w∈A} ψ̃_{w} = π that ∑_{w∈A_{0}} τ_{w} ≥ m_{0}, which completes the proof.

3 Accumulating procedure

To describe the accumulating procedure we need some information about metric structure of the universal covering X of (M, g), where g is a geometrization.

3.1 Metric structure of the universal covering

Recall (see, for example, [BS], [CK]) that X can be represented as the countable union X = ∪_{v}X_{v} of blocks, where each X_{v} is a closed convex subset in X isometric to the metric product F_{v} × ℝ, and F_{v} is the universal covering of a compact hyperbolic surface S_{v} with geodesic boundary. Every two blocks are either disjoint or intersect over a boundary component which is a 2-flat in X separating them and consequently no three blocks have a point in common. The 2-flats in X, which separate blocks, are called the walls. A wall w common for blocks X_{v}, X_{v′} covers a 2-torus e ⊂ M, which separates (may be locally) the blocks M_{v} = π(X_{v}), M_{v′} = π(X_{v′}) of M. The metric decompositions X_{v} = F_{v} × ℝ, X_{v′} = F_{v′} × ℝ do not agree on w, and their ℝ-factors induce two fibrations of w by parallel geodesics. We denote by α_{w} the angle between these fibrations, 0 < α_{w} ≤ π/2. Since M is compact and the set E of separating tori in M is finite, we have α_{0} := inf_{w} α_{w} > 0, where the infimum is taken over all the walls in X.

3.2 Modified Poincaré series

We fix a wall w^{∗} ⊂ X, take a block X_{v^{∗}} ⊂ X, for which w^{∗} is a boundary wall, and take a base point x_{0} ∈ w^{∗}. We denote by W_{0} the set of the boundary walls of X_{v^{∗}} different from w^{∗}; by W_{n}, n ≥ 1 the set of walls in X at the combinatorial distance n + 1 from w^{∗}, that is, w ∈ W_{n} if and only if any geodesic segment in X between x_{0} and w intersets n walls over its interior, including a wall from W_{0}. Note that W = ∪_{n≥0}W_{n} consists of all walls in X, which lie on one and the same side of w^{∗} as X_{v^{∗}}. Now, we define a modified Poincaré series as follows

P_{W} (t) = ∑_{w∈W} e^{−t dist(x_{0},w)}.

Comparing P_{W} with P(t) = ∑_{γ∈Γ} e^{−t|x_{0}−γx_{0}|}, one easily obtains from triangle inequality that P(t) ≥ e^{−D}P_{W} (t), where D > 0 is the maximal diameter of the tori e ∈ E. Recall that the critical exponent of P is defined as the infimum of t ∈ ℝ for which P(t) < ∞. Therefore, the critical exponent \overline{h} of P_{W} satisfies \overline{h} ≤ h(g), and to prove Theorem 1.1 it suffices to show that \overline{h} > 1.

3.3 Special broken-geodesic paths

Proposition 3.1. For each n ≥ 0 we have

P_{n}(1) := ∑_{w∈W_{n}} e^{− dist(x_{0},w)} ≥ \frac{π}{4} λ_{0}^{n},

where λ_{0} > 1 is the constant from Lemma 2.1.

Proof. Induction over n. For each n ≥ 0 and each wall w ∈ W_{n} we produce a broken geodesic ξ_{w} in X between x_{0} and w as follows. For w ∈ W_{0} we put ξ_{w} = x_{0}x_{w}, where x_{w} ∈ w is the point closest to x_{0}. Note that x_{0}x_{w} lies in a horizontal slice F_{v^{∗}} × {r_{0}} of the block X_{v^{∗}} = F_{v^{∗}} × ℝ.

Assume that for all k, 0 ≤ k ≤ n − 1 and all w ∈ W_{k} the broken geodesic ξ_{w} is already defined, ξ_{w} is a juxtaposition η_{w_{0}}η_{w_{1}} … η′_{w_{k}} of geodesic segments with the sequence w_{i} ∈ W_{i} of walls leading to w = w_{k}. Moreover, we assume that each segment of ξ_{w} lies in a block and connects its different boundary components, the last segment η′_{w_{k}} lies in a horizontal slice of the block, which contains it, and η′_{w_{k}} is orthogonal to the wall w.

Take w ∈ W_{n}. Then there is a unique \overline{w} ∈ W_{n−1} which precedes w. By the assumption, the last edge η′ ; \overline{w} = s\overline{w}x\overline{w} of ξ\overline{w} lies in a horizontal slice F_{v′} × \{r_{v′}\} of its block X_{v′} = F_{v′} × ℝ and it is orthogonal to the wall \overline{w} (at the end point x\overline{w}). The important feature of our construction is that ξ_{w} contains all segments of ξ\overline{w} but the last one η′ ; \overline{w}.

Let X_{v} ⊂ X be the other block adjacent to \overline{w}, in particular, the walls w, \overline{w} are its boundary components. Recall that the metric splitting X_{v} = F_{v}×ℝ does not agree with that of X_{v′} along of \overline{w}. Let F_{v} × \{r_{v}\} be the horizontal slice of X_{v} which contains x\overline{w} on the corresponding boundary component. The boundary lines of the slices F_{v′} ×\{r_{v′}\}, F_{v} ×\{r_{v}\}, which lie in \overline{w}, contain x\overline{w} and form an angle α\overline{w} ∈ (α_{0}, π/2].

To construct ξ_{w} we do the following. We take an isometric copy F_{v} ⊂ H^{2} ×\{r_{v′}\} of F_{v} × \{r_{v}\} (by rotating the last by the angle α\overline{w}), where F_{v′} × {r_{v′}} ⊂ H^{2} ×{r_{v′}}, so that F_{v′} × {r_{v′}} and F_{v} are sitting in the hyperbolic plane H^{2} ×{r_{v′}} and they are adjacent along the common boundary component, for which we use the same notation \overline{w}. Now, we connect the initial point s\overline{w} of the last edge η′ ; \overline{w} ⊂ ξ\overline{w} with the boundary component of F_{v} corresponding to w by the shortest geodesic segment s\overline{w}x′_{w} ⊂ H^{2} ×\{r_{v′}\} and take t_{w} = s\overline{w}x′_{w} ∩ \overline{w}. The segment t_{w}x′_{w} turned back to F_{v} × \{r_{v}\} gives the last segment η′_{w} = s_{w}x_{w} of ξ_{w}. So s_{w} ∈ \overline{w} ∩ F_{v} × \{r_{v}\}, |s_{w} − x\overline{w}| = |t_{w} − x\overline{w}| and η′_{w} ⊂ F_{v} × {r_{v}} is orthogonal to w at x_{w}. To complete the construction of ξ_{w}, we delete the last segment η′ ; \overline{w} ⊂ ξ\overline{w} replacing it by η\overline{w}η′_{w}, where η\overline{w} = s\overline{w}s_{w} ⊂ X_{v′}. Clearly, so constructed ξ_{w} has all properties advertised above.

Let l = |s\overline{w} − x\overline{w}| = L(η′ ; \overline{w}) be the length of the last segment of ξ\overline{w}, l′_{w} = L(η\overline{w}), l′′_{w} = L(η′_{w}). Then we have

L(ξ_{w}) = L(ξ\overline{w}) − l + l′_{w} + l′′_{w} = L(ξ\overline{w}) − l + l̃_{w} − Δ_{w},

where l̃_{w} = L(s\overline{w}x′_{w}), l̃′_{w} = L(s\overline{w}t_{w}) so that l̃_{w} = l̃′_{w} + l′′_{w}, and Δ_{w} = l̃′_{w} − l′_{w}. We obtained the same configuration which was studied in sect. 2, and we are going to apply Lemma 2.1 to estimate P_{n}(1) from below. Note that l being the length of a segment in X connecting different boundary components of a block is separated from 0 by some positive constant l_{0}, which depends only on M, l ≥ l_{0} > 0.

For n = 0 we have

P_{0}(1) = ∑_{w∈W_{0}} e^{−|x_{0}−x_{w}|} = ∑_{w∈W_{0}} tan \frac{ψ_{w}}{4} ≥ \frac{π}{4} ,

where ψ_{w} is the angle under which the boundary component w of X_{v^{∗}} is observed from x_{0} (in the horizontal direction).

By inductive assumption we have

P_{n−1}(1) ≥ ∑_{w∈W_{n−1}} e^{−L(ξ_{w})} ≥ \frac{π}{4} λ_{0}^{n−1}.

Represent W_{n} = ∪\overline{w}W_{n,\overline{w}}, where the union is taken over all \overline{w} ∈ W_{n−1} and each w ∈ W_{n,\overline{w}} follows \overline{w}. Then

P_{n}(1) ≥ ∑_{w∈W_{n}} e^{−L(ξ_{w})} = ∑_{\overline{w}∈W_{n−1}} e^{−L(ξ\overline{w})}e^{l} ∑_{w∈W_{n,\overline{w}}} e^{Δ_{w}}e^{−l̃_{w}}.

Applying Lemma 2.1 with A = W_{n,\overline{w}}, we obtain

P_{n}(1) ≥ λ_{0} ∑_{\overline{w}∈W_{n−1}} e^{−L(ξ\overline{w})} ≥ \frac{π}{4} λ_{0}^{n},

which completes the proof.

4 Self-similarity argument

The constant λ_{0} > 1 from Proposition 3.1 depends only on some metric data of M. Thus there is n ∈ ℕ, n = n(M), so that \frac{π}{4} λ_{0}^{n} > 1. It follows from Proposition 3.1 that

P_{n}(\overline{h}) = ∑_{w∈W_{n}} e^{−\overline{h} dist(x_{0},w)} ≥ 1

for some \overline{h} > 1. Furthermore, taking n sufficiently large, we can find \overline{h} > 1 with P_{n}(\overline{h}) ≥ 1 for any choice of the initial block X_{v^{∗}}, its wall w^{∗} and the base point x_{0} ∈ w^{∗}, since the set of choices up to isometries of X is compact.

We fix n ∈ ℕ with this property and select a subset W^{∗} ⊂ W, W^{∗} = ∪_{k≥1}W_{k}^{∗}, where W_{k}^{∗} = W_{kn}. The set W_{1}^{∗} serves as the generating set for W^{∗}. Connecting x_{0} with each wall w ∈ W_{1}^{∗} by the shortest geodesic segment x_{0}x_{w}, we obtain new base points x_{w} ∈ w (these x_{w} may be different from x_{w} constructed in the proof of Proposition 3.1). By induction, we find a base point x_{w} ∈ w for each w ∈ W_{k}^{∗}, k ≥ 1 with the property dist(x\overline{w}, w) = |x\overline{w} − x_{w}|, where \overline{w} ∈ W_{k−1}^{∗} precedes w. Furthermore, by the choice of n and \overline{h} we have

∑_{w} e^{−\overline{h}|x\overline{w}−x_{w}|} ≥ 1,

for each \overline{w} ∈ W_{k−1}^{∗}, where the summation is taken over all w ∈ W_{k}^{∗} which follow \overline{w}. Since dist(x_{0}, w) ≤ dist(x_{0}, x_{w}) ≤ dist(x_{0}, x\overline{w}) + |x\overline{w} − x_{w}|, we obtain

P_{W_{k}^{∗}}(\overline{h}) = ∑_{w∈W_{k}^{∗}} e^{−\overline{h} dist(x_{0},w)}
≥ ∑_{w∈W_{k−1}^{∗}} e^{−\overline{h} dist(x_{0},x_{w})} ≥ ∑_{w∈W_{n}} e^{−\overline{h} dist(x_{0},x_{w})} ≥ 1

for each k ≥ 1. Therefore, the modified Poincaré series

P_{W} (\overline{h}) ≥ P_{W^{∗}}(\overline{h}) = ∑_{k≥1} P_{W_{k}^{∗}}(\overline{h})

diverges at \overline{h} and hence h(g) ≥ \overline{h} > 1. This completes the proof of Theorem 1.1.

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